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Mathematically Strong Subystems of Analysis with Low Rate of Growth Of Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals Ulrich Kohlenbach Fachbereich Mathematik J.W. Goethe–Universit¨at D–60054 Frankfurt, Germany September 1995 Abstract This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A ω hierarchy (GnA )n∈IN of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. ω We establish the following extraction rule for an extension of GnA by quantifier–free choice δ η AC–qf and analytical axioms Γ having the form ∀x ∃y ≤ρ sx∀z F0 (including also a ‘non– standard’ axiom F − which does not hold in the full set–theoretic model but in the strongly majorizable functionals): ω 1 0 0 From a proof GnA +AC–qf + Γ ⊢∀u ,k ∀v ≤τ tuk∃w A0(u,k,v,w) one can extract a uniform bound Φ such that 1 0 ∀u ,k ∀v ≤τ tuk∃w ≤ ΦukA0(u,k,v,w) holds in the full set–theoretic type structure. In case n = 2 (resp. n = 3) Φuk is a polynomial (resp. an elementary recursive function) in M ω − k, u := λx. max(u0,...,ux). In the present paper we show that for n ≥ 2, GnA +AC–qf+F proves a generalization of the binary K¨onig’s lemma yielding new conservation results since the ω conclusion of the above rule can be verified in Gmax(3,n)A in this case. In a subsequent paper we will show that many important ineffective analytical principles ω and theorems can be proved already in G2A +AC–qf+Γ for suitable Γ. 1 Introduction This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. 0 Let U be a complete separable metric space, K a compact metric space and A ∈ Σ1. As we have elaborated in [21] many numerically interesting theorems in analysis can be transformed into sentences having the form (1) ∀u ∈ U, k ∈ IN∀v ∈ K∃w ∈ IN A(u,k,v,w) 1 and one is interested in a uniform bound Φuk on w which does not depend on v ∈ K, i.e. ∀u ∈ U, k ∈ IN∀v ∈ K∃w ≤ Φuk A(u,k,v,w). Quite often A is monotone with respect to w, i.e. A(u,k,v,w1) ∧ w2 ≥ w1 → A(u,k,v,w2) and hence the bound Φuk in fact realizes ‘∃w’ (see [21] for a discussion of this phenomenon). What do we know about the rate of growth of Φ if we know that (1) is proved using certain parts of analysis? In [14],[15], [19],[20] we have developed a proof–theoretic method suited for the extraction of such bounds from proofs in analysis which guarantees the extractability of primitive recursive bounds for large parts of analysis. Moreover this method has been applied to concrete (ineffective) proofs in approximation theory yielding new a–priori estimates for numerically relevant data as constants of strong unicity and others which improve known estimates significantly (see [19],[20],[21]). In analyzing these applications we developed in [21] a new monotone functional interpretation which has important advantages over the method from [15] and provides a particular perspicuous procedure of analyzing ineffective proofs in analysis. The starting point for the investigation carried out in the present paper is the following problem: Whereas the general meta–theorems in [15], [19] and [21] only guarantee the existence of a primitive recursive bound Φ, the bounds which are actually obtained in our applications to approximation theory have a very low rate of growth which is polynomial (of degree ≤ 2) relatively to the growth of the data of the problem. Thus the problem arises to close the still large gap between polynomial and primitive recursive growth. Before we start to discuss this question let us note that using a suitable representation of spaces like U,X and the basic notions of real analysis, sentences (1) can be formalized in the language of arithmetic in all finite types such that (1) gets (a special case of) the following logical form1 1 0 0 (2) ∀u , k ∀v ≤τ tu k∃w A0(u, k,v,w). 1 1 1 0 0 0 Here u := u1,...,un, k := k1,...,km, t is a closed term, τ an arbitrary finite type, 1 = 0(0) and A0(u, k,v,w) a quantifier–free formula containing only the free variables u, k,v,w. ≤τ is defined pointwise. By a uniform bound we now mean a functional Φ such that 1 0 ∀u , k ∀v ≤τ tu k∃w ≤0 Φu kA0(u, k,v,w). Again the predicate ‘uniform’ for the bound Φ refers to the fact that Φ does not depend on v. Coming back to our question from above we are interested in the determination of those parts of classical analysis, where the extractability of bounds Φ having only polynomial growth (resp. elementary recursive growth) relatively to the data is guaranteed. ω In order to address this question we introduce a hierarchy GnA of subsystems of classical arith- metic in all finite types and investigate the rate of growth caused by various analytical principles ω 1(1) ω relatively to GnA +AC–qf. The definable functionals t in GnA are of increasing order of growth: 1 ω For the weak system G2A discussed below more subtle representations than those which are used in [19] are necessary. Such representations are developed in 3 of [22] and will be published in a paper under preparation. 2 If n = 1, then tf 1x0 is bounded by a linear function in f M , x, if n = 2, then tf 1x0 is bounded by a polynomial in f M , x, if n = 3, then tf 1x0 is bounded by an elementary recursive (i.e. a (fixed) finitely iterated exponential) function in f M , x, where f M := λx0. max(f0,...,fx) and Φfx is called linear (polynomial, elementary recursive) in 1 0 0 0 1 1 f, x if ∀f , x (Φfx =0 Φ[f, x]) for a term Φ[f, x] which is built up from 0 , x ,f ,S , + (respectively 00, x0,f 1,S 1, +, · and 00, x0,f1,S1, +, ·, λx0,y0.xy) only. In our results the term Φ[f, x] can always be constructed. Let us motivate this notion for the polynomial case: If Φfx is a polynomial in f 1, x0, then in particular for every polynomial p ∈ IN[x] the function λx0.Φpx can be written as a polynomial in IN[x]. Moreover there exists a polynomial q ∈ IN[x] (depending only on the term structure of Φ) such that For every polynomial p ∈ IN[x] one can construct a polynomial r ∈ IN[x] such that 1 0 ∀f f ≤1 p → ∀x (Φfx ≤0 r(x)) and deg(r) ≤ q(deg(p)). 1(1) ω M M Since every closed term t in G2A is bounded by a polynomial Φf x in f , x and f ≤1 p → M f ≤1 p (since p is monotone) this also holds for tfx instead of Φfx. k 1 0 (0) . (0) ω In particular every closed term t (t ) of G2A is bounded by a polynomial pt ∈ IN[x] (resp. a polynomial pt ∈ IN[x1,...,xk]). 1 ω n For general n ∈ IN, n ≥ 1, every closed term t of GnA is bounded by some function ft ∈E where En denotes the n–th level of the Grzegorczyk hierarchy. ω It turns out that many basic concepts of real analysis can be defined already in G2A : e.g. rational numbers, real numbers (with their usual arithmetical operations and inequality rela- tions), d–tuples of real numbers (for every fixed d), sequences and series of reals, continuous functions F : IRd → IR and uniformly continuous functions F : [a,b]d → IR, the supremum of F ∈ C([a,b]d, IR) on [a,b]d, the Riemann integral of F ∈ C[a,b]. Furthermore the trigonometric functions sin, cos, tan, arcsin, arccos, arctan and π as well as the restriction expk (lnk) of the expo- ω nential function (logarithm) to [−k, k] for every fixed number k can be introduced in G2A (The ω unrestricted functions exp and ln can be defined in G3A ). ω G2A +AC–qf proves many of the basic properties of these objects. − − In this paper we determine the growth of extactable bounds Φ for GnA+AC–qf+F , where F ω is a certain analytical axiom which allows (relatively to G2A +AC–qf) very short and perspicuous proofs of fundamental theorems of analysis as e.g. • every pointwise continuous function f : [0, 1]d → IR is uniformly continuous and possesses a modulus of uniform continuity • the attainment of the maximum value of f ∈ C([0, 1]d, IR) on [0, 1]d • the sequential form of the Heine–Borel covering property for [0, 1]d 3 • Dini’s theorem together with a modulus of uniform convergence • the existence of a uniformly continuous inverse function for every strictly increasing continuous function f : [0, 1] → IR. In particular we show the following: δ τ Let ∆ be a set of sentences having the form ∀x ∃y ≤ρ sx∀z B0 (B0 quantifier–free). Then the following rule holds: ω − 1 0 0 From a given proof GnA +AC–qf+∆ + F ⊢ ∀u , k ∀v ≤τ tu k∃w A0(u, k,v,w) (∗) one can extract a uniform bound Φ such that G Aω + ∆+ b-AC ⊢ ∀u1, k0∀v ≤ tu k∃w ≤ Φu k A (u, k,v,w), max(n,3) i τ 0 0 where Φu k is a polynomial in uM , k if n =2 Φu k is an elementary recursive function in uM , k if n =3.
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